reserve V for Z_Module;
reserve W, W1, W2 for Submodule of V;

theorem XXTh1:
  for V be free Z_Module, I, A be finite Subset of V
  st I is Basis of V & card (I) + 1 = card (A)
  holds A is linearly-dependent
  proof
    let V be free Z_Module, I, A be finite Subset of V;
    assume AS: I is Basis of V & card (I) + 1 = card (A);
    reconsider IQ =(MorphsZQ(V)).:(I) as Subset of Z_MQ_VectSp(V);
    reconsider IQ as finite Subset of Z_MQ_VectSp(V);
    P2:IQ is Basis of Z_MQ_VectSp(V) by AS,ThEQRZMV3D;
    assume P31: not A is linearly-dependent;
    reconsider B = (MorphsZQ(V)).:(A) as Subset of Z_MQ_VectSp(V);
    reconsider B as finite Subset of Z_MQ_VectSp(V);
    PP: IQ is linearly-independent & Lin (IQ) = the ModuleStr of Z_MQ_VectSp(V)
    by VECTSP_7:def 3,P2;
    B is linearly-independent by P31,ThEQRZMV3C;
    then P5: card(B) <= card(IQ) by VECTSP_9:19,PP;
    card(IQ) = card(I) by ThEQRZMV3E;
    then card(A) <= card(I) by P5,ThEQRZMV3E;
    hence contradiction by AS,NAT_1:13;
  end;
